
The cosmological constant as the dark energy candidate characterized by its equation of state \(w_{\rm DE} = -1\) .

The dark energy with a constant equation of state other than the cosmological constant (i.e. \(w_{\rm DE} \ne -1\) ).Let us comment that from now on we shall use \(w\)  instead of \(w_{\rm DE}\)  and we label the corresponding cosmological model as \(w\) CDM (instead of \(w_{\rm DE}\) CDM) as already done in the literature.

The case when the dark energy equation of state is dynamical and it assumes the Chevallier-Polarski-Linder parametrization [1]}, [2]}:
\(w_{\rm DE} = w_0 + w_a (1-a)~,\) 
where \(w_0\)  is the present value of \(w_{\rm DE}\)  and \(w_a = dw_{\rm DE}/da\)  at \(a = 1\)  is another free parameter in this parametrization.Note that the corresponding cosmological model is labeled as \(w_0w_a\) CDM.

Finally, we consider a very exotic dark energy model, namely the phenomenologically emergent dark energy (PEDE) where DE does not have any effective presence in the past but it emerges at late times. In this model scenario, the dark energy density (\(\rho _{\rm DE}\) ) is directly parametrized which takes the form [3]}, [4]}
\(\frac{\rho _{\rm DE}}{\rho _{\rm c,0}} = \Omega _{\rm DE,0} \Bigl [1- \tanh (\log _{10} (1+z)) \Bigr ]~,\) 
where \(\rho _{\rm c, 0}\)  is the critical energy density at present time and \(\Omega _{\rm DE,0}\)  is the present value of the DE density parameter \(\Omega _{\rm DE} \equiv \rho _{\rm DE}/\rho _{\rm crit}\) . We note that even if this parametrization has a dynamical nature, this does not offer any additional degree of freedom, or free parameter.

